In this lecture, we study Monte Carlo simulations for the model of interacting fermions known as the Hubbard model, for particles with spin 1/2. The story that we will develop is how one of the more complicated models of interacting quantum particles turns out to be ... simply a slightly more involved-than-usual classical Ising model

Hubbard model - hamiltonian

$H = - t \sum_{\langle i,j\rangle}c_i^+ c_j + \text{h.c.} + U \sum_i n_i^{\uparrow} n_i ^{\downarrow}$

The Trotter formula for fermions

Hirsch decoupling (Gaussian decoupling formula for a binary variable)

These days, Jorge Hirsch is uniformally famous for a bibliometric index, the h-index. For computational physicists, he is the author of the first simulations on the Hubbard model, of the famous Hirsch decoupling, and of the incredible Hirsch-Fye algorithm. Here we describe the decoupling.

In the following table, we set
$X = \exp(-\Delta \tau U n^{\uparrow} n^{\downarrow})$
, and we compute the action of the operator X on the four states with
$n^{\uparrow} n^{\downarrow} = \pm 1$

$\begin{array}{c|cc|c} X&0&1& n^{\uparrow}\\ \hline 0&1&1\\ 1&1& \exp (-\Delta \tau U) \\ n^{\downarrow}\\ \end{array}$

In the following table, we set
$Y = \frac{1}{2}\left[ \exp(\lambda [ n^{\uparrow}- n^{\downarrow} ] -\frac{\Delta \tau}{2} U [n^{\uparrow} + n^{\downarrow}] ) + \exp(-\lambda [ n^{\uparrow}- n^{\downarrow} ] -\frac{\Delta \tau}{2} U [n^{\uparrow} + n^{\downarrow}] ) \right]$
,
and likewise compute the action of the operator X on the four states with
$n^{\uparrow} n^{\downarrow} = \pm 1$

$\begin{array}{c|cc|c} Y&0&1& n^{\uparrow}\\ \hline 0&1&\exp(-\frac{\Delta \tau}{2} U) \left[\frac{\exp (\lambda)}{\exp( -\lambda)}\right]\\ 1&1& \exp (-\Delta \tau U) \\ n^{\downarrow}\\ \end{array}$

which leads to the condition
$\cosh \lambda = \exp(-\frac{\Delta \tau}{2} U)$

and to the final result:
$\exp(-\Delta \tau U n^{\uparrow} n^{\downarrow}) = \text{Tr}_{\sigma = \pm 1} \exp\left[ \lambda \sigma (n^{\uparrow} - n^{\downarrow}) - \frac{\Delta \tau}{2} U (n^{\uparrow} + n^{\downarrow}) \right]$

The trace in this expression is the sum over the values +/- 1 of a simple Ising variable.

The Blankenbecler-Scalapino-Sugar (BSS) determinant formula

The point of the BSS formula is that the fermion trace over the exponential
of a bilinear operator expression can be done easily.
$\text{Tr} \left[ \exp (-c_i A_{ij}c_j) \exp( - c_i B_{ij} c_j\right] = = \det \left[ 1 + \exp(-A) \exp(-B) \right]$

This formula can be derived using Grassmann algebras. A direct calculation was provided by J. E. Hirsch in the appendix of <ref name="Hirsch"> J. E. Hirsch ''Two-dimensional Hubbard model: Numerical simulation study'' Physical Review B 31, 4403 </ref>:

$Z = \sum_{\text{states} \alpha } \langle \alpha |\exp(- \beta H) | \alpha \rangle$

We first show how to treat bilinear terms, such as

$\text{Tr} \exp( - c_i^+ B_{ij} c_j) = \text{Tr} \prod_\mu \exp( - c_\mu^+ b_\mu c_\mu)$

For this simple case, we can expand the exponential into

$\exp( - c_\mu^+ b_\mu c_\mu) = 1 - c_\mu^+ b_\mu c_\mu + \frac{1}{2}c_\mu^+ b_\mu c_\mu c_\mu^+ b_\mu c_\mu - \frac{1}{3!}.....$

Using
$c_\mu^+ c_\mu + c_\mu c_\mu^+ = 1$
we reach

$\exp( - c_\mu^+ b_\mu c_\mu) = 1 [[[+ c_\mu^+ b_\mu c_\mu ]]] + \frac{1}{2}c_\mu^+ b_\mu^2 c_\mu - \frac{1}{3!}c_\mu^+ b_\mu^3 c_\mu .... [[[-c_\mu^+ b_\mu c_\mu ]]] = \prod_\mu [ 1 + ( \exp (- b_\mu) - 1) c_\mu^+ c_\mu$

This expression is easily evaluated. It yields the expression
$\det[1 + \exp(-B)]$

More generally, <ref name="Hirsch"/>, one finds:

$\text{Tr} \exp( - c_i^+ A_{ij} c_j)\exp( - c_i^+ B_{ij} c_j) = \det[1 + \exp(-A) \exp(-B) ]$

This non-trivial formula is the key to determinantal fermion methods. We note that in it, the trace is over the fermion variables, that is we have integrated out the quantum variables. Notice that, at a difference with what we did in the last lecture, the fermion trace is over all occupations of fermions: we must introduce a chemical potential to fix the density of electrons.

Fermion algorithm

In the following, we put all pieces together. For simplicity, we imagine a one-dimensional Hubbard model:

$H = -t \sum_i c_i^+ c_j + U \sum_i n_i^{\uparrow} n_i^{\downarrow} = H_0 + H_1$

$\exp( -\beta H) = \prod \exp( -\tau H) \exp( -\tau H) \exp( -\tau H) \exp( -\tau H) ...$

Using the Hirsch decoupling discussed just above, we have that the partition function is

$Z = \sum_{\sigma_1 = \pm 1} .... \sum_{\sigma_L = \pm 1} \left[ \exp (-\tau H_0) \exp(\lambda \sigma_1 (n^{\uparrow} - n^{\downarrow} - \frac{\tau}{2} U (n^{\uparrow} + n^{\downarrow}) \right] \times \times \times$
We can now separate the (quantum) spin up from the (quantum) spin down and arrive, at each slice, arrive at an expression in terms of a hopping matrix K which connects nearest neighbors and a diagonal matrix V, which depends on the Ising spin.

$\exp ( - c_i^{+ \uparrow} (\tau K_{ij}) c_i^{ \uparrow}) \exp ( - c_i^{+ \uparrow} V(\sigma) c_j )$

As we can use the BSS formula for performing the trace over the fermions,
we are left with

$Z = \text{Tr}_{\sigma} \prod_{\alpha = \pm 1} \det [1 + B_L(\alpha) B_{L-1}(\alpha) \cdots B_1(\alpha)] = \text{Tr}_{\sigma} \det O_{\uparrow} O_{\downarrow}$

We have reached the final point of our discussion, namely the representation of an interacting fermion problem in terms of an Ising model with a strange interaction, namely a product of two determinants. This expression is due to Blankenbecler, Scalapino and Sugar, and the first simulations of the Hubbard model were done by Hirsch. Both are magnificent achievements in theoretical physics. Much of the complexity has still been hidden, for example concerning the calculation of Greens functions and other observables.

References